Dividing Fractions

Almost everyone is taught the same three words for this — keep, change, flip — and almost nobody is told why they work. That is a shame, because the reason is simpler than the rule and it makes the rule impossible to get backwards.

Start with what division actually asks: how many of these fit into that? Twelve divided by 3 is 4 because four 3s fit into 12. Fractions are no different, and you can count them.

Count the pieces, and the flip appears

The bar is one whole. The pale shading is the first fraction — what you are dividing up. The numbered blocks are the second fraction, laid along it end to end, and the numbers are the count.

At the starting values you are asking how many eighths fit into three quarters. Six blocks appear, so:

(3/4)/(1/8) = 6

Now look at what counting eighths is. There are 8 of them in every whole, so asking how many eighths something contains is asking how many times bigger than an eighth it is — and that is multiplying by 8:

(3/4)/(1/8) = 3/4 * 8/1 = 24/4 = 6

That is the flip. Dividing by a fraction with 8 underneath means multiplying by 8, because there are 8 of them in every unit. Turning 1/8 into 8/1 is not a trick applied to the numbers; it is the question rewritten.

The flipped fraction has a name, the reciprocal, and the rule in full is: dividing by a fraction is multiplying by its reciprocal.

Only the second one flips

Drag the sliders so the second fraction is larger than the first and the count falls below 1 — the last block is dashed, because only part of it fits. That partial block is why the answer is a fraction rather than a whole number, and it is also a good check on which fraction you flipped.

Division is not symmetric, so the order matters:

(3/4)/(1/2) = 3/4 * 2/1 = 3/2

(1/2)/(3/4) = 1/2 * 4/3 = 2/3

Same two fractions, different answers. If you flip the wrong one you get the reciprocal of the right answer, which is a mistake that hides well — both look like plausible fractions. Saying the calculation out loud fixes it: the number you are dividing by is the one that flips.

Whole numbers, both ways round

A whole number is a fraction over 1, so dividing by 5 means multiplying by 1/5:

(2/3)/5 = 2/3 * 1/5 = 2/15

which is what sharing two thirds between five people should give — something small. The other direction is worth doing too:

6/(1/3) = 6 * 3/1 = 18

Eighteen thirds in six wholes. Whenever the answer to a division comes out bigger than what you started with, check whether you divided by something below 1 — if you did, a bigger answer is correct, not a slip.

A mixed number becomes improper first

2 + 1/2 divided by 3/4 is not "2 divided by something". Write the mixed number as 5/2 and then nothing is special about it:

(5/2)/(3/4) = 5/2 * 4/3 = 20/6 = 10/3

Try one

(2/3)/(4/9)

If you got 8/27, the first fraction was flipped instead of the second. If you got 2/3 times 4/9, nothing was flipped at all — the change from division to multiplication and the flip always happen together.